Q. If x i s real,then `(x^2+2x+c)/(x^2+4x+3c)` can take all real values if A) `0<c<2` B) `-1<c<1` C) `0<=c<=1` D) none of these

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Given `(x^2+2x+c)/(x^2+4x+3c)` , determine the value(s) of c so that the expression can take all real values:

(A) Assuming the question means that you can put in any real value for x:

Suppose c=0: `(x^2+2x)/(x^2+4x)=(x(x+2))/(x(x+4))` . x cannot be zero so B and C are incorrect.

Suppose c=1: `(x^2+2x+1)/(x^2+4x+3)=((x+1)^2)/((x+1)(x+3))` . x cannot be -1 so A is incorrect.

The answer is D -- none of the above.

(B) Assuming the question meant that the expression could take on all real values (the output could be any real number) the answer is also D.

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